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<h1 class="mume-header" id="%E5%9B%BE%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C%E5%AF%BC%E8%AE%BA">图神经网络导论</h1>

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<h2 class="mume-header" id="%E5%9B%BE%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C">图神经网络</h2>

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<h4 class="mume-header" id="tengzhanghusteducn"><a href="mailto:tengzhang@hust.edu.cn">tengzhang@hust.edu.cn</a></h4>

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<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>编码器 - 解码器模型</h5></div></div>
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<p>编码器：表示学习</p>
<ul>
<li>网格型数据，可采用卷积神经网络</li>
<li>序列型数据，可采用循环神经网络</li>
</ul>
<br>
<p>对于一般的图数据，编码器该如何设计？</p>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="47" class="slide " data-line="47" data-h="1" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>编码器设计</h5></div></div>
<p>输入：<span class="mathjax-exps">$\Gcal = (\Vcal, \Ecal)$</span>，<span class="mathjax-exps">$\Xv = [\xv_1, \ldots, \xv_{|\Vcal|}] \in \Rbb^{d \times |\Vcal|}$</span>，边的特征，整图的特征</p>
<br>
<p>一个 naïve 的做法是将邻接矩阵<span class="mathjax-exps">$\Av$</span>拉直成向量，后接一个全连接网络</p>
<p>

$$
\begin{align*}
    \zv_\Gcal = \mlp ([\Av]_{:,1} \oplus [\Av]_{:,2} \oplus \cdots \oplus [\Av]_{:,|\Vcal|})
\end{align*}
$$
</p>

<p>邻接矩阵人为给点指定了顺序，点和边自带的特征没有用起来</p>
<p>引入置换矩阵<span class="mathjax-exps">$\Pv$</span>，定义</p>
<ul>
<li>平移不变性：<span class="mathjax-exps">$f(\Pv \Av \Pv^\top) = f(\Av)$</span>，<span class="blue">标量值函数</span>对于点的标号顺序不敏感</li>
<li>平移等价性：<span class="mathjax-exps">$\fv (\Pv \Av \Pv^\top) = \Pv \fv (\Av)$</span>，<span class="blue">向量值函数</span>与点的标号顺序始终一致</li>
</ul>
<br>
<p>神经消息传递 (neural message passing) 框架：GNN 的统一表示</p>
<ul>
<li>点之间不断进行信息交互</li>
<li>信息通过神经网络进行更新</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section></section><section><section data-notes="" lineno="83" class="slide " data-line="83" data-h="2" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>消息传递框架</h5></div></div>
<p>记第<span class="mathjax-exps">$k$</span>轮点<span class="mathjax-exps">$u$</span>的隐藏状态 (嵌入表示) 为<span class="mathjax-exps">$\hv_u^\sup{k}$</span>，GNN 的消息传递形式化为</p>
<p>

$$
\begin{align*}
    \hv_u^\sup{0}   &amp; = \xv_u, ~ \forall u \in \Vcal                                                                                         \\
    \hv_u^\sup{k+1} &amp; = \update^\sup{k} \left( \hv_u^\sup{k}, \aggregate^\sup{k} ( \{ \hv_v^\sup{k} \mid \forall v \in \Ncal(u) \} ) \right) \\
                    &amp; = \update^\sup{k} \left( \hv_u^\sup{k}, \mv_{\Ncal(u)}^\sup{k} \right)                                                 \\
    \zv_u           &amp; = \hv_u^\sup{K}, ~ \forall u \in \Vcal
\end{align*}
$$
</p>

<ul>
<li><span class="mathjax-exps">$\update (\cdot)$</span>：用来更新当前点隐藏状态的可微函数</li>
<li><span class="mathjax-exps">$\aggregate (\cdot)$</span>：用来汇聚邻居信息的可微函数，其输入是一个集合 (平移等价性)</li>
<li><span class="mathjax-exps">$\mv_{\Ncal(u)}^\sup{k}$</span>：点<span class="mathjax-exps">$u$</span>的所有邻居的信息汇聚结果</li>
<li>如果点本身没有特征，可以用点层面的统计量作为特征</li>
<li>GNN 一层可视为一轮消息传递，<span class="mathjax-exps">$K$</span>层即编码了所有<span class="mathjax-exps">$K$</span>阶邻居的信息</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="110" class="slide " data-line="110" data-h="2" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>消息传递框架</h5></div></div>
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" class="center width80 top10">
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section></section><section><section data-notes="" lineno="118" class="slide " data-line="118" data-h="3" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>GNN 的基本形式</h5></div></div>
<p>最早版本的 GNN：</p>
<p>

$$
\begin{align*}
    \hv_u^\sup{k} = \sigma \left( \Wv_\self^\sup{k} ~ \hv_u^\sup{k-1} + \Wv_\neigh^\sup{k} \sum_{v \in \Ncal(u)} \hv_v^\sup{k-1} + \bv^\sup{k} \right)
\end{align*}
$$
</p>

<ul>
<li><span class="mathjax-exps">$\Wv_\self^\sup{k}$</span>、<span class="mathjax-exps">$\Wv_\neigh^\sup{k}$</span>、<span class="mathjax-exps">$\bv^\sup{k}$</span>是可训练的参数，偏置<span class="mathjax-exps">$\bv^\sup{k}$</span>通常可以省略</li>
<li><span class="mathjax-exps">$\sigma$</span>是逐元素的非线性变换，例如之前介绍过的各种激活函数</li>
</ul>
<br>
<p>用消息传递框架表示：</p>
<ul>
<li><span class="mathjax-exps">$\aggregate ( \{ \hv_v \mid \forall v \in \Ncal(u) \} ) = \mv_{\Ncal(u)} = \class{blue}{\sum_{v \in \Ncal(u)} \hv_v}$</span>，注意求和是点序无关的</li>
<li><span class="mathjax-exps">$\update ( \hv_u, \mv_{\Ncal(u)} ) = \sigma ( \Wv_\self ~ \hv_u + \Wv_\neigh ~ \mv_{\Ncal(u)} + \bv)$</span></li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="146" class="slide " data-line="146" data-h="3" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>GNN 的基本形式</h5></div></div>
<p>记<span class="mathjax-exps">$\Hv^\sup{k} = [\hv_1^\sup{k}, \ldots, \hv_{|\Vcal|}^\sup{k}]$</span></p>
<p>GNN 的基本形式的矩阵表示：</p>
<p>

$$
\begin{align*}
    \Hv^\sup{k} = \sigma ( \Wv_\self^\sup{k} ~ \Hv^\sup{k-1} + \Wv_\neigh^\sup{k} \Hv^\sup{k-1} \Av )
\end{align*}
$$
</p>

<br>
<p>添加自指向环，将显式的<span class="mathjax-exps">$\update(\cdot)$</span>隐藏于<span class="mathjax-exps">$\aggregate(\cdot)$</span>之中</p>
<p>

$$
\begin{align*}
    \hv_u^\sup{k} &amp; = \aggregate^\sup{k} ( \{ \hv_v^\sup{k} \mid \forall v \in \Ncal(u) \class{blue}{\cup \{u\}} \} ) \\
    \Hv^\sup{k} &amp; = \sigma (\Wv^\sup{k} \Hv^\sup{k-1} (\Av \class{blue}{+ \Iv}))
\end{align*}
$$
</p>

<ul>
<li>优点：共享<span class="mathjax-exps">$\Wv_\self$</span>和<span class="mathjax-exps">$\Wv_\neigh$</span>，可以避免过拟合</li>
<li>缺点：<span class="mathjax-exps">$\aggregate(\cdot)$</span>要求平移等价性，这样无法区分自身的信息和邻居的信息</li>
</ul>
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</section></section><section data-notes="" lineno="184" class="slide " data-line="184" data-h="4" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>GNN 的各种魔改</h5></div></div>
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</p><div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section><section data-notes="" lineno="192" class="slide " data-line="192" data-h="5" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>归一化</h5></div></div>
<p>GNN 基本形式采用直接加和汇聚邻居的信息</p>
<br>
<p>问题：对度数敏感，若<span class="mathjax-exps">$d_u \gg d_v$</span>，则<span class="mathjax-exps">$\| \sum_{w \in \Ncal(u)} \hv_w \| \gg \| \sum_{w \in \Ncal(v)} \hv_w \|$</span>，<span class="mathjax-exps">$u$</span>的信息可以“淹没”其它点的信息</p>
<br>
<p>方案：归一化</p>
<p>

$$
\begin{align*}
    \mv_{\Ncal(u)} = \frac{\sum_{v \in \Ncal(u)} \hv_v}{|\Ncal(u)|}, \quad \mv_{\Ncal(u)} = \sum_{v \in \Ncal(u)} \frac{\hv_v}{\sqrt{|\Ncal(u)||\Ncal(v)|}}
\end{align*}
$$
</p>

<p>后者与自指向环结合：图卷积网络 (<strong>g</strong>raph <strong>c</strong>onvolutional <strong>n</strong>etwork, GCN)</p>
<p>

$$
\begin{align*}
    \hv_u^\sup{k} = \sigma \left( \Wv^\sup{k} \sum_{ v \in \Ncal(u) \cup \{u\} } \frac{\hv_v}{\sqrt{|\Ncal(u)||\Ncal(v)|}} \right)
\end{align*}
$$
</p>

<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="230" class="slide " data-line="230" data-h="5" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>集合汇聚</h5></div></div>
<p>归一化 → 具有置换不变性的集合函数</p>
<br>
<p>集合汇聚 (set pooling)</p>
<p>

$$
\begin{align*}
    \mv_{\Ncal(u)} = \mlp_\thetav \left( \sum_{v \in \Ncal(u)} \mlp_\phiv (\hv_v) \right)
\end{align*}
$$
</p>

<p>理论上可以证明任意具有置换不变性的集合函数都可由上式无限逼近</p>
<br>
<p>Janossy 汇聚：通过考虑所有的标号排列获得置换不变性</p>
<p>

$$
\begin{align*}
    \mv_{\Ncal(u)} = \mlp_\thetav \left( \frac{1}{|\Pi|} \sum_{\pi \in \Pi} \rho_\phi (\hv_{v_1}, \hv_{v_2}, \ldots, \hv_{v_{|\Ncal(u)|}})_{\pi_i} \right)
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\Pi$</span>是由置换构成的集合，<span class="mathjax-exps">$\rho_\phi$</span>是任意函数 (不需要具有置换不变性)</p>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="270" class="slide " data-line="270" data-h="5" data-v="2">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>邻居注意力</h5></div></div>
<p>为邻居赋予不同的权重：<span class="mathjax-exps">$\mv_{\Ncal(u)} = \sum_{v \in \Ncal(u)} \alpha_{u,v} \hv_v$</span></p>
<br>
<p>图注意力网络 (<strong>g</strong>raph <strong>at</strong>tention network, GAT)</p>
<p>

$$
\begin{align*}
    \alpha_{u,v} = \frac{\exp(\av^\top [\Wv \hv_u \oplus \Wv \hv_v])}{\sum_{v' \in \Ncal(u)} \exp(\av^\top [\Wv \hv_u \oplus \Wv \hv_{v'}])}
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\av$</span>和<span class="mathjax-exps">$\Wv$</span>是可训练的参数，<span class="mathjax-exps">$\oplus$</span>表示向量拼接</p>
<br>
<p>变种：</p>
<p>

$$
\begin{align*}
    \alpha_{u,v} = \frac{\exp(\av^\top [\hv_u^\top \Wv \hv_v])}{\sum_{v' \in \Ncal(u)} \exp(\av^\top [\hv_u^\top \Wv \hv_{v'}])}, ~ \alpha_{u,v} = \frac{\exp(\mlp(\hv_u, \hv_v))}{\sum_{v' \in \Ncal(u)} \exp(\mlp(\hv_u, \hv_{v'}))}
\end{align*}
$$
</p>

<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section></section><section><section data-notes="" lineno="308" class="slide " data-line="308" data-h="6" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>更新</h5></div></div>
<p>对更新进行魔改主要为了解决 GNN 的过平滑 (over-smoothing) 问题</p>
<br>
<p>过平滑：当 GNN 的层数 (迭代轮数) 过大时，所有点的信息趋于相同</p>
<br>
<p>设<span class="mathjax-exps">$f: \Rbb_+ \mapsto \Rbb_+$</span>为可微的归一化函数，对任意带有自指向环并采用</p>
<p>

$$
\begin{align*}
    \aggregate ( \{ \hv_v, \forall v \in \Ncal(u) \cup \{u\} \} ) = \frac{1}{f (|\Ncal(u) \cup \{u\}|)} \sum_{v \in \Ncal(u) \cup \{u\}} \hv_v
\end{align*}
$$
</p>

<p>进行信息汇聚的 GNN 有<span class="mathjax-exps">$\ev^\top (\partial \hv_v^\sup{k} / \partial \hv_u^\sup{0}) \ev \propto p_K (u | v)$</span></p>
<br>
<p>当<span class="mathjax-exps">$K$</span>过大时，<span class="mathjax-exps">$p_K (u | v)$</span>即为随机游走的平稳分布，这意味着每个点的局部邻居信息已经完全丢失</p>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="340" class="slide " data-line="340" data-h="6" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>跳连</h5></div></div>
<p>基本想法：类似于残差网络，引入跳连 (skip connection) 凸显点自身的信息，防止被邻居信息“淹没”</p>
<br>
<p>拼接跳连：初见于 GraphSAGE 算法</p>
<p>

$$
\begin{align*}
    \update_{\text{concat}} (\hv_u, \mv_{\Ncal(u)}) = [\update_{\text{base}} (\hv_u, \mv_{\Ncal(u)}) \oplus \hv_u]
\end{align*}
$$
</p>

<p>插值跳连：</p>
<p>

$$
\begin{align*}
    \update_{\text{interpolate}} (\hv_u, \mv_{\Ncal(u)}) = \alphav_1 \odot \update_{\text{base}} (\hv_u, \mv_{\Ncal(u)}) + \alphav_2 \odot \hv_u
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\alphav_1, \alphav_2 \in [0,1]^d$</span>为门向量 (gating vectors)，<span class="mathjax-exps">$\alphav_2 = \ev - \alphav_1$</span>，<span class="mathjax-exps">$\alphav_1$</span>与 GNN 原本的参数联合学习</p>
<ul>
<li><span class="mathjax-exps">$\alphav_1$</span>可以作为一个单层 GNN 的输出</li>
<li><span class="mathjax-exps">$\alphav_1$</span>也可以作为一个 MLP 的输出</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="379" class="slide " data-line="379" data-h="6" data-v="2">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>门机制</h5></div></div>
<p>对于点<span class="mathjax-exps">$u$</span>：</p>
<ul>
<li>每轮汇聚得到的邻居信息<span class="mathjax-exps">$\mv_{\Ncal(u)}^\sup{k}$</span>构成一个序列，可以看作 RNN 的输入<span class="mathjax-exps">$\xv^\sup{k}$</span></li>
<li>每轮更新得到的嵌入表示<span class="mathjax-exps">$\hv_u^\sup{k}$</span>构成一个序列，可以看作 RNN 的隐藏状态</li>
</ul>
<p>

$$
\begin{align*}
    \hv_u^\sup{k} &amp; = \gru (\hv_u^\sup{k-1}, \mv_{\Ncal(u)}^\sup{k}) \\
    \hv_u^\sup{k} &amp; = \lstm (\hv_u^\sup{k-1}, \mv_{\Ncal(u)}^\sup{k})
\end{align*}
$$
</p>

<p>事实上任何 RNN 都可引入到 GNN 中来，不局限于 GRU 和 LSTM</p>
<br>
<p>权值共享</p>
<ul>
<li>在每个点上训练一个 RNN，不同点之间权值共享</li>
<li>通常丹师们还会将 GNN 的汇聚层也进行权值共享</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="410" class="slide " data-line="410" data-h="6" data-v="3">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>全层拼接</h5></div></div>
<p>之前我们都是假设将 GNN 的最后一层当作点的嵌入表示<span class="mathjax-exps">$\zv_u = \hv_u^\sup{k}$</span>，但层数一深最后一层各点的嵌入表示趋于相同 (过平滑)</p>
<br>
<p>拼接所有层的嵌入表示，称为 <strong>j</strong>umping <strong>k</strong>nowledge (JK) connections</p>
<p>

$$
\begin{align*}
    \zv_u = f_{\text{JK}} (\hv_u^\sup{0} \oplus \hv_u^\sup{1} \oplus \cdots \oplus \hv_u^\sup{K})
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$f_{\text{JK}}(\cdot)$</span>是任意可微函数：</p>
<ul>
<li>恒等函数</li>
<li>max-pooling</li>
<li>LSTM</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section></section><section><section data-notes="" lineno="438" class="slide " data-line="438" data-h="7" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>异构图</h5></div></div>
<p>关系图卷积网络 (<strong>r</strong>elational <strong>g</strong>raph <strong>c</strong>onvolutional <strong>n</strong>etwork, RGCN)</p>
<p>

$$
\begin{align*}
    \mv_{\Ncal(u)} = \sum_{\tau \in \Rcal} \sum_{v \in \Ncal_\tau (u)} \frac{\Wv_\tau \hv_v}{f_n(\Ncal(u), \Ncal(v))}
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$f_n(\cdot, \cdot)$</span>是归一化函数，之前介绍过的归一化函数都可用过来</p>
<br>
<p>权值共享：引入一组基矩阵<span class="mathjax-exps">$\Bv_1, \ldots, \Bv_b$</span></p>
<p>

$$
\begin{align*}
    \mv_{\Ncal(u)} = \sum_{\tau \in \Rcal} \sum_{v \in \Ncal_\tau (u)} \frac{\sum_{i \in [b]} \alpha_{i, \tau} \Bv_i \hv_v }{f_n(\Ncal(u), \Ncal(v))}
\end{align*}
$$
</p>

<p>扩展</p>
<ul>
<li>不带权值共享</li>
<li>加入注意力机制</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="477" class="slide " data-line="477" data-h="7" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>加权图</h5></div></div>
<p>RGCN 假设边的的特征是离散的，因此汇聚可以采用直接求和的形式</p>
<br>
<p>对于更一般形式的边的特征，可以和邻居的嵌入表示进行拼接</p>
<p>

$$
\begin{align*}
    \mv_{\Ncal(u)} = \aggregate_\base ( \{ \hv_v \oplus \ev_{(u, \tau, v)}, \forall v \in \Ncal(u) \} )
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\ev_{(u, \tau, v)}$</span>是边<span class="mathjax-exps">$(u, \tau, v)$</span>的向量值特征</p>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section></section><section><section data-notes="" lineno="501" class="slide " data-line="501" data-h="8" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>图汇聚</h5></div></div>
<p>对于图层面的任务，还需整图的嵌入表示<span class="mathjax-exps">$\zv_\Gcal$</span></p>
<br>
<p>集合汇聚策略：学习汇聚函数<span class="mathjax-exps">$f_{\mathrm{p}}: \{ \zv_1, \ldots, \zv_{|\Vcal|} \} \mapsto \zv_\Gcal$</span></p>
<p>

$$
\begin{align*}
    \zv_\Gcal = \frac{\sum_{u \in \Vcal} \zv_u}{f_n(|\Vcal|)}
\end{align*}
$$
</p>

<br>
<p>事实上之前介绍的任意对邻居的汇聚函数都可用过来</p>
<p>套用 LSTM 和注意力机制，初始化<span class="mathjax-exps">$\qv_0$</span>、<span class="mathjax-exps">$\ov_0$</span>为全零向量，对<span class="mathjax-exps">$t \in [T]$</span>有</p>
<p>

$$
\begin{align*}
    \qv_t = \lstm(\ov_{t−1}, \qv_{t−1}), ~ \alpha_{v,t} &amp; = \frac{\exp(f_a (\zv_t, \qv_t))}{\sum_{u \in \Vcal} \exp(f_a (\zv_u, \qv_t))}, ~ \ov_t = \sum_{v \in \Vcal} \alpha_{v,t} \zv_v
\end{align*}
$$
</p>

<p>最终输出<span class="mathjax-exps">$\zv_\Gcal = \ov_1 \oplus \ov_2 \oplus \cdots \oplus \ov_T$</span></p>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="541" class="slide " data-line="541" data-h="8" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>图汇聚</h5></div></div>
<p>集合汇聚没有考虑图的结构信息</p>
<br>
<p>图粗化 (coarsening) 策略：构建层次化的 GNN，类似于 CNN 中的汇聚</p>
<ul>
<li>假设有聚类函数<span class="mathjax-exps">$\fv_c: \Gcal \times \Rbb^{|\Vcal| \times d} \mapsto \Rbb^{|\Vcal| \times c}$</span>将图中的点分成<span class="mathjax-exps">$c$</span>个簇</li>
<li>记<span class="mathjax-exps">$\Sv = \fv_c(\Gcal, \Zv)$</span>为分配矩阵，<span class="mathjax-exps">$[\Sv]_{u,i}$</span>表示点<span class="mathjax-exps">$u$</span>和第<span class="mathjax-exps">$i$</span>个簇之间的相关性</li>
<li><span class="mathjax-exps">$f_c$</span>简单的可采用谱聚类，复杂的也可再用一个 GNN</li>
</ul>
<br>
<p>以簇为点构造新的粗粒度的图，对应的邻接矩阵和点特征矩阵：</p>
<p>

$$
\begin{align*}
    \Av_\new = \Sv^\top \Av \Sv \in \Rbb^{c \times c}, \quad \Xv_\new = \Sv^\top \Xv \in \Rbb^{c \times d}
\end{align*}
$$
</p>

<ul>
<li>如果新图粒度已然合适，就在新图上再运行一个 GNN 得到各点 (原图中的簇) 的嵌入表示，然后用集合汇聚得到整图的嵌入表示；否则继续聚类</li>
<li>如果想整个模型是端到端的，那么聚类函数<span class="mathjax-exps">$f_c$</span>必须是可微的</li>
<li>如果想改善模型的计算复杂度和训练速度，可采用删点而不是汇聚</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section></section><section data-notes="" lineno="575" class="slide " data-line="575" data-h="9" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>一般形式的消息传递</h5></div></div>
<p>之前介绍的汇聚-更新基本模式可以进一步一般化</p>
<br>
<p>引入边和图的嵌入表示</p>
<p>

$$
\begin{align*}
    \hv_{(u,v)}^\sup{k} &amp; = \class{blue}{\update_\edge} (\hv_{(u,v)}^\sup{k-1}, \hv_u^\sup{k-1}, \hv_v^\sup{k-1}, \hv_\Gcal^\sup{k-1}) \\
    \mv_{\Ncal(u)} &amp; = \aggregate_\node (\{ \hv_{(u,v)}^\sup{k}, \forall v \in \Ncal(u) \}) \\
    \hv_u^\sup{k} &amp; = \update_\node (\hv_u^\sup{k-1}, \mv_{\Ncal(u)}, \class{blue}{\hv_\Gcal^\sup{k-1}} ) \\
    \hv_\Gcal^\sup{k} &amp; = \class{blue}{\update_\graph} (\hv_\Gcal^\sup{k-1}, \{ \hv_u^\sup{k}, \forall u \in \Vcal \}, \{ \hv_{(u,v)}^\sup{k}, \forall (u,v) \in \Ecal \})
\end{align*}
$$
</p>

<ul>
<li>每轮的更新顺序是先边、再点，最后整图</li>
<li>点更新引入了整图的嵌入表示，即同时考虑局部和全局信息</li>
<li>整图更新可以采用前面介绍的汇聚、粗化等策略</li>
<li>近期有研究表明一般形式的消息传递更适合学习逻辑表达式</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section data-notes="" lineno="605" class="slide " data-line="605" data-h="10" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>GNN 实现 点分类</h5></div></div>
<p>设编码器给出的点的嵌入表示为<span class="mathjax-exps">$\zv_u$</span>，类别标记<span class="mathjax-exps">$\yv_u \in \{ 0,1 \}^C$</span></p>
<br>
<p>解码器的损失函数通常采用负对数似然</p>
<p>

$$
\begin{align*}
    \Lcal = - \sum_{u \in \Vcal_\train} \log \left( \sum_{c \in [C]} [\yv_u]_c \frac{\exp(\wv_c^\top \zv_u)}{\sum_{i \in [C]} \exp(\wv_i^\top \zv_u)} \right)
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\wv_1, \ldots, \wv_C$</span>是待学习的参数</p>
<br>
<p>两类学习任务</p>
<ul>
<li>转导学习：待预测的点在训练时已知，参与消息传递，不参与损失函数计算</li>
<li>归纳学习：待预测的点及相关的边在训练时未知，既不参与消息传递，也不参与损失函数计算，通常用于有多个连通分支的图，在一个连通分支上训练，在另一个连通分支上预测</li>
</ul>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section data-notes="" lineno="636" class="slide " data-line="636" data-h="11" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>图层面的任务</h5></div></div>
<p>设图的嵌入表示为<span class="mathjax-exps">$\zv_\Gcal$</span></p>
<br>
<p>分类，训练数据为<span class="mathjax-exps">$(\Gcal_1, \yv_1), \ldots, (\Gcal_m, \yv_m)$</span>，类别标记<span class="mathjax-exps">$\yv_j \in \{ 0,1 \}^C$</span>，损失采用负对数似然</p>
<p>

$$
\begin{align*}
    \Lcal = - \sum_{j \in [m]} \log \left( \sum_{c \in [C]} [\yv_j]_c \frac{\exp(\wv_c^\top \zv_{\Gcal_j})}{\sum_{i \in [C]} \exp(\wv_i^\top \zv_{\Gcal_j})} \right)
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\wv_1, \ldots, \wv_C$</span>是待学习的参数</p>
<br>
<p>回归，训练数据为<span class="mathjax-exps">$(\Gcal_1, y_1), \ldots, (\Gcal_m, y_m)$</span>，损失采用平方损失函数</p>
<p>

$$
\begin{align*}
    \Lcal = - \sum_{i \in [m]} \| \mlp(\zv_{\Gcal_i}) - y_{\Gcal_i} \|_2^2
\end{align*}
$$
</p>

<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section><section data-notes="" lineno="674" class="slide " data-line="674" data-h="12" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>关系预测</h5></div></div>
<p>编码器给出的点的嵌入表示为<span class="mathjax-exps">$\zv_u$</span></p>
<br>
<p>关系预测的问题形式化：</p>
<p>

$$
\begin{align*}
    \Lcal = \sum_{(u,v) \in \Dcal} l(\dec(\zv_u, \zv_v), [\Sv]_{uv})
\end{align*}
$$
</p>

<p>其中</p>
<ul>
<li><span class="mathjax-exps">$\Dcal$</span>是训练点对集合</li>
<li><span class="mathjax-exps">$l(\cdot, \cdot)$</span>是损失函数</li>
<li><span class="mathjax-exps">$\dec(\cdot, \cdot)$</span>是解码器</li>
<li><span class="mathjax-exps">$[\Sv]_{uv}$</span>描述点<span class="mathjax-exps">$u$</span>、<span class="mathjax-exps">$v$</span>间的关系</li>
</ul>
<p>后三者在选择上可以排列组合</p>
<div class="footer"><hr class="hr_bottom"><div class="multi_column"><h6 class="bottom_left">图神经网络导论</h6><h6 class="bottom_center">图神经网络</h6><h6 class="bottom_right">tengzhang@hust.edu.cn</h6></div></div>
</section><section vertical="true" data-notes="" lineno="705" class="slide " data-line="705" data-h="12" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>基于矩阵分解的方法</h5></div></div>
<p>拉普拉斯特征映射：</p>
<ul>
<li><span class="mathjax-exps">$\dec(\zv_u, \zv_v) = \| \zv_u - \zv_v \|_2^2$</span></li>
<li><span class="mathjax-exps">$\Lcal = \sum_{(u,v) \in \Dcal} \dec (\zv_u, \zv_v) ~ [\Sv]_{uv}$</span></li>
</ul>
<p>即相似的点要有相似的嵌入表示，若<span class="mathjax-exps">$\Sv$</span>为拉普拉斯矩阵，<span class="mathjax-exps">$\zv_u \in \Rbb^d$</span>，则最优解就是<span class="mathjax-exps">$\Sv$</span>的最小<span class="mathjax-exps">$d$</span>个特征向量，事实上就是在做谱聚类</p>
<br>
<p>内积法：</p>
<ul>
<li><span class="mathjax-exps">$\dec(\zv_u, \zv_v) = \zv_u^\top \zv_v$</span></li>
<li><span class="mathjax-exps">$\Lcal = \sum_{(u,v) \in \Dcal} \| \dec (\zv_u, \zv_v) - [\Sv]_{uv} \|_2^2$</span></li>
</ul>
<p>代表性方法有 Graph Factorization (GF)、GraRep、HOPE，区别仅在于采用的<span class="mathjax-exps">$\Sv$</span>不同，GF 直接用<span class="mathjax-exps">$\Sv = \Av$</span>，GraRep 用<span class="mathjax-exps">$\Av$</span>的幂次，HOPE 可用之前介绍的任意两点间的局部重合度量</p>
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</section><section vertical="true" data-notes="" lineno="727" class="slide " data-line="727" data-h="12" data-v="2">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>基于随机游走的方法</h5></div></div>
<p>基于矩阵分解的方法均采用固定的<span class="mathjax-exps">$\Sv$</span>，多为邻接矩阵<span class="mathjax-exps">$\Av$</span>的多项式函数</p>
<p>采用随机、非对称的<span class="mathjax-exps">$\Sv$</span>：</p>
<p>

$$
\begin{align*}
    \dec(\zv_u, \zv_v) &amp; = \frac{\exp((\Wv_{\cen} \zv_u)^\top (\Wv_{\con} \zv_v))}{\sum_{k \in \Vcal} \exp((\Wv_{\cen} \zv_u)^\top (\Wv_{\con} \zv_k))} \approx P_T (v|u) \\
    \Lcal &amp; = - \sum_{(u,v) \in \Dcal} \log ( \dec(\zv_u, \zv_v) )
\end{align*}
$$
</p>

<ul>
<li><span class="mathjax-exps">$P_T (v|u)$</span>为从点<span class="mathjax-exps">$u$</span>开始进行<span class="mathjax-exps">$T$</span>步随机游走访问到<span class="mathjax-exps">$v$</span>的概率</li>
<li><span class="mathjax-exps">$\Dcal$</span>是从<span class="mathjax-exps">$u$</span>开始随机游走序列构成的集合</li>
</ul>
<p>损失函数涉及 Softmax 操作，计算复杂度<span class="mathjax-exps">$O(|\Dcal||\Vcal|)$</span></p>
<ul>
<li>DeepWalk 采用结构化 Softmax 将计算复杂度降低到<span class="mathjax-exps">$O(|\Dcal|\log|\Vcal|)$</span></li>
<li>node2vec 添加了负采样<span class="mathjax-exps">$\Lcal = - \sum_{(u,v) \in \Dcal} \log ( \sigma (\zv_u^\top \zv_v) ) - \gamma \Ebb_{w \sim P(\Vcal)} [\log (-\sigma (\zv_u^\top \zv_w) )]$</span></li>
</ul>
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</section><section vertical="true" data-notes="" lineno="756" class="slide " data-line="756" data-h="12" data-v="3">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>关系预测</h5></div></div>
<p>问题形式化：</p>
<p>

$$
\begin{align*}
    \Lcal = \sum_{(u,v) \in \Dcal} l(\dec(\zv_u, \zv_v), [\Sv]_{uv})
\end{align*}
$$
</p>

<p>方法总结：</p>
<div class="threelines">
<table>
<thead>
<tr>
<th style="text-align:center">方法</th>
<th style="text-align:center">解码器</th>
<th style="text-align:center">相似性度量</th>
<th style="text-align:center">损失函数</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">Lap. eig.</td>
<td style="text-align:center"><span class="mathjax-exps">$\| \zv_u - \zv_v \|^2$</span></td>
<td style="text-align:center">随意</td>
<td style="text-align:center"><span class="mathjax-exps">$\dec(\zv_u, \zv_v) \cdot [\Sv]_{uv}$</span></td>
</tr>
<tr>
<td style="text-align:center">Graph Fact.</td>
<td style="text-align:center"><span class="mathjax-exps">$\zv_u^\top \zv_v$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$[\Av]_{uv}$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$\| \dec (\zv_u, \zv_v) - [\Sv]_{uv} \|_2^2$</span></td>
</tr>
<tr>
<td style="text-align:center">GraRep</td>
<td style="text-align:center"><span class="mathjax-exps">$\zv_u^\top \zv_v$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$[\Av]_{uv}, \ldots, [\Av^k]_{uv}$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$\| \dec (\zv_u, \zv_v) - [\Sv]_{uv} \|_2^2$</span></td>
</tr>
<tr>
<td style="text-align:center">HOPE</td>
<td style="text-align:center"><span class="mathjax-exps">$\zv_u^\top \zv_v$</span></td>
<td style="text-align:center">随意</td>
<td style="text-align:center"><span class="mathjax-exps">$\| \dec (\zv_u, \zv_v) - [\Sv]_{uv} \|_2^2$</span></td>
</tr>
<tr>
<td style="text-align:center">DeepWalk</td>
<td style="text-align:center"><span class="mathjax-exps">$\exp(\zv_u^\top \zv_v) / \sum_{k \in \Vcal} \exp(\zv_u^\top \zv_k)$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$P(v \mid u)$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$- [\Sv]_{uv} \log (\dec(\zv_u, \zv_v))$</span></td>
</tr>
<tr>
<td style="text-align:center">node2vec</td>
<td style="text-align:center"><span class="mathjax-exps">$\exp(\zv_u^\top \zv_v) / \sum_{k \in \Vcal} \exp(\zv_u^\top \zv_k)$</span></td>
<td style="text-align:center">有偏<span class="mathjax-exps">$P(v \mid u)$</span></td>
<td style="text-align:center"><span class="mathjax-exps">$- [\Sv]_{uv} \log (\dec(\zv_u, \zv_v))$</span></td>
</tr>
</tbody>
</table>
</div>
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</section></section><section><section data-notes="" lineno="789" class="slide " data-line="789" data-h="13" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>多重关系预测</h5></div></div>
<p>常见任务：知识图谱补全</p>
<br>
<p>此时解码器为<span class="mathjax-exps">$\dec: \Rbb^d \times \Rcal \times \Rbb^d \mapsto \Rbb_+$</span>，<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v)$</span>可看作边<span class="mathjax-exps">$(u, \tau, v)$</span>存在于图中的似然</p>
<br>
<p>多重关系预测的问题形式化：</p>
<p>

$$
\begin{align*}
    \Lcal = \sum_{(u,\tau,v) \in \Ecal} l(\dec(\zv_u, \tau, \zv_v), [\Av]_{u,\tau,v})
\end{align*}
$$
</p>

<p>其中<span class="mathjax-exps">$\Av \in \Rbb^{|\Vcal| \times |\Rcal| \times |\Vcal|}$</span>是邻接张量</p>
<br>
<p>对于多重关系预测</p>
<ul>
<li>相似性度量通常就是邻接张量，因为高阶的邻接张量没有物理意义</li>
<li>损失函数和解码器依然可以排列组合</li>
</ul>
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</section><section vertical="true" data-notes="" lineno="824" class="slide " data-line="824" data-h="13" data-v="1">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>多重关系预测</h5></div></div>
<p>非多重关系预测中使用的损失函数<span class="mathjax-exps">$\| \dec (\zv_u, \zv_v) - [\Sv]_{uv} \|_2^2$</span>这里不太适合</p>
<ul>
<li>非多重关系预测中，<span class="mathjax-exps">$\Sv$</span>可以为<span class="mathjax-exps">$\Av$</span>的幂次，多值更适合看作回归问题</li>
<li>多重关系预测中，<span class="mathjax-exps">$\Sv$</span>就是邻接张量，二值更适合看作分类问题</li>
</ul>
<p>带负采样的交叉熵：</p>
<p>

$$
\begin{align*}
    \Lcal = - \sum_{(u,\tau,v) \in \Ecal} \log (\dec(\zv_u, \tau, \zv_v)) - \gamma \Ebb_{w \sim P_u(\Vcal)} [\log (\sigma (-\dec(\zv_u, \tau, \zv_w))]
\end{align*}
$$
</p>

<ul>
<li>前者是正确预测“存在边”的对数似然</li>
<li>后者是正确预测“不存在边”的对数似然，<span class="mathjax-exps">$P_u(\Vcal)$</span>是跟点<span class="mathjax-exps">$u$</span>相关的概率分布</li>
</ul>
<p>Hinge 损失：</p>
<p>

$$
\begin{align*}
    \Lcal = \sum_{(u,\tau,v) \in \Ecal} \sum_{w \in P_u(\Vcal)} \max \{0, - \dec(\zv_u, \tau, \zv_v) + \dec(\zv_u, \tau, \zv_w) + \Delta \}
\end{align*}
$$
</p>

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</section><section vertical="true" data-notes="" lineno="862" class="slide " data-line="862" data-h="13" data-v="2">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>多重关系预测</h5></div></div>
<p>解码器品种繁多，可以和前面两种损失任意搭配</p>
<br>
<p>RESCAL：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = \zv_u^\top \Rv_\tau \zv_v$</span>，每种关系对应一个<span class="mathjax-exps">$\Rv_\tau \in \Rbb^{d \times d}$</span></p>
<br>
<p>平移解码</p>
<ul>
<li>TransE：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = - \| \zv_u + \rv_\tau - \zv_v \|$</span></li>
<li>TransX：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = - \| g_{1,\tau} (\zv_u) + \rv_\tau - g_{2,\tau}(\zv_v) \|$</span></li>
<li>TransH：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = - \| (\zv_u - \wv_\tau^\top \zv_u \wv_\tau ) + \rv_\tau - (\zv_v - \wv_\tau^\top \zv_v \wv_\tau ) \|$</span></li>
</ul>
<br>
<p>多重线性点积 DistMult：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = \sum_{i \in [d]} [\zv_u]_i [\rv_\tau]_i [\zv_v]_i$</span>，有对称性</p>
<br>
<p>复数解码</p>
<ul>
<li>ComplEx：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = \Re (\sum_{i \in [d]} [\zv_u]_i [\rv_\tau]_i \overline{[\zv_v]_i})$</span>，其中<span class="mathjax-exps">$\zv_u, \tau, \zv_v \in \Cbb^d$</span></li>
<li>RotatE：<span class="mathjax-exps">$\dec(\zv_u, \tau, \zv_v) = - \| \zv_u \odot \rv_\tau - \zv_v \|$</span>，其中<span class="mathjax-exps">$[\rv_\tau]_i = \exp(j ~ \theta_{\tau, i})$</span></li>
</ul>
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</section><section vertical="true" data-notes="" lineno="893" class="slide " data-line="893" data-h="13" data-v="3">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>多重关系预测</h5></div></div>
<p>可以通过对逻辑关系的表示考察解码器的能力</p>
<ul>
<li>对称性：<span class="mathjax-exps">$(u, \tau, v) \in \Ecal \leftrightarrow (v, \tau, u) \in \Ecal$</span></li>
<li>反对称性：<span class="mathjax-exps">$(u, \tau, v) \in \Ecal \leftrightarrow (v, \tau, u) \not \in \Ecal$</span></li>
</ul>
<br>
<p>DistMult 解码器仅具有对称性，TransE 仅具有反对称性：</p>
<p>

$$
\begin{align*}
    - \| \zv_u + \rv_\tau - \zv_v \| = - \| \zv_v + \rv_\tau - \zv_u \| \Rightarrow \rv_\tau^\top (\zv_u  - \zv_v) = 0 \Rightarrow \rv_\tau = \zerov
\end{align*}
$$
</p>

<ul>
<li>逆性：<span class="mathjax-exps">$(u, \tau_1, v) \in \Ecal \leftrightarrow (v, \tau_2, u) \in \Ecal$</span></li>
<li>复合性：<span class="mathjax-exps">$(u, \tau_1, w) \in \Ecal \wedge (w, \tau_2, v) \in \Ecal \rightarrow (v, \tau_3, u) \in \Ecal$</span></li>
</ul>
<br>
<p>TransE 具有复合性：</p>
<p>

$$
\begin{align*}
    - \| \zv_u + \rv_{\tau_1} + \rv_{\tau_2} - \zv_v \| = - \| \underbrace{\zv_u + \rv_{\tau_1} - \zv_w}_{=\zerov} + \underbrace{\zv_w + \rv_{\tau_2} - \zv_v}_{=\zerov} \|
\end{align*}
$$
</p>

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</section></section><section data-notes="" lineno="935" class="slide " data-line="935" data-h="14" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>预训练 GNN</h5></div></div>
<p>以最小化邻居重构损失预训练的 GNN 和随机初始化的 GNN 性能相仿</p>
<br>
<p>Deep Graph Infomax (DGI)</p>
<p>

$$
\begin{align*}
    \Lcal = - \sum_{u \in \Vcal_\train} ( \Ebb_\Gcal \log(\overbrace{D(\zv_u, \zv_\Gcal)}^{\nearrow}) + \gamma \Ebb_{\widetilde{\Gcal}} \log (1 - \overbrace{D(\widetilde{\zv}_u, \zv_\Gcal)}^{\searrow}) )
\end{align*}
$$
</p>

<ul>
<li><span class="mathjax-exps">$\widetilde{\Gcal}$</span>：原图<span class="mathjax-exps">$\Gcal$</span>的随机扰动版</li>
<li><span class="mathjax-exps">$\zv_u$</span>：点<span class="mathjax-exps">$u$</span>的嵌入表示</li>
<li><span class="mathjax-exps">$\widetilde{\zv}_u$</span>：点<span class="mathjax-exps">$u$</span>在扰动图<span class="mathjax-exps">$\widetilde{\Gcal}$</span>上的嵌入表示</li>
<li><span class="mathjax-exps">$D(\cdot, \cdot)$</span>：判别当前的嵌入表示来自原图还是扰动图，通常是个神经网络</li>
</ul>
<br>
<p>基本思想：GNN 生成的点嵌入表示在原图和扰动图上应该显著不同</p>
<br>
<p>DGI 也可用于监督学习设定下的辅助损失</p>
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</section><section data-notes="" lineno="970" class="slide " data-line="970" data-h="15" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>效率问题</h5></div></div>
<p>各个点独自做消息传播会产生大量重复计算，可采用矩阵形式计算</p>
<p>

$$
\begin{align*}
    \Hv^\sup{k} = \sigma ( \Wv_\self^\sup{k} ~ \Hv^\sup{k-1} + \Wv_\neigh^\sup{k} \Hv^\sup{k-1} \Av )
\end{align*}
$$
</p>

<ul>
<li>整图操作非常耗内存</li>
<li>反向传播训练时也是用 GD，而不是 SGD</li>
</ul>
<br>
<p>小批量：随机采样一些点可能得不到一个连通子图，无法做消息传播</p>
<br>
<p>方案：</p>
<ul>
<li>先确定一小部分点，递归地加入这些点的邻居，直到得到连通图</li>
<li>设置每个点的邻居个数上界，避免产生太大的子图</li>
</ul>
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</section><section data-notes="" lineno="1002" class="slide " data-line="1002" data-h="16" data-v="0">
<div class="header"><img class="hust" src=""><div class="title"><hr class="hr_top"><h5>正则化</h5></div></div>
<p>适用于 GNN 的传统正则化</p>
<ul>
<li><span class="mathjax-exps">$\ell_2$</span>惩罚</li>
<li>随机丢弃</li>
<li>层归一化</li>
</ul>
<br>
<p>各层权值共享</p>
<ul>
<li>所有<span class="mathjax-exps">$\aggregate (\cdot)$</span>和<span class="mathjax-exps">$\update (\cdot)$</span>均共享权值</li>
<li>适用于超过 6 层的 GNN</li>
</ul>
<br>
<p>边随机丢弃：于 GAT 中首次提出</p>
<ul>
<li>训练时随机删除一些边</li>
<li>适用于知识图谱</li>
</ul>
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